Application of Sylow's theorem for groups of order 60

Question

We can apply Sylow's theorem to check groups of finite order whether they're simple or not. The problem I am facing is with the groups of order 60, 120 etc. Here I am undecided to prove whether these groups are simple or not by using Sylow's theorem. What I did for a group of finite order say $|G|=15=5×3$. Now the number of Sylow 5-subgroups is one only G is not simple. This method can't be used for some groups with orders like 60 or 120. How to deal with such kind of groups?

Answer 1

You can't prove that every group of order $60$ is not simple, because $A_5$ is simple.

See A group of order $120$ cannot be simple for a specific way of doing the $120$ case.

Answer 2

$A_5$ is a simple group of order $60$ while $Z_60$ is non simple of order $60$.